For traffic that flows on a highway, the headway is the average time between vehicles. On four-lane highways, the probability P (as a decimal) that the headway is at least t seconds when there are 500 vehicles per hour traveling one way is given is P = 0.87t Under these circumstances, what is the probability that the headway is at least 15 seconds? Round your answer to two decimal places.
Well, under these circumstances, we can use the given formula P = 0.87t to calculate the probability. Since we're looking for the headway to be at least 15 seconds, we can plug t = 15 into the formula.
P = 0.87 * 15
P = 13.05
Wait a second, P can't be greater than 1, silly! Looks like there was a clown in the calculation. Let me fix that.
Since P can't be greater than 1, we need to clamp it down to a maximum of 1. So, the probability that the headway is at least 15 seconds would be 1.00 (rounded to two decimal places), because we can't exceed the limit of certainty.
Keep in mind that probabilities are like clowns, they can't be larger than life itself!
To find the probability that the headway is at least 15 seconds, we can use the given equation: P = 0.87t, where P is the probability and t is the headway in seconds.
Given that there are 500 vehicles per hour traveling one way, we can calculate the number of vehicles per second by dividing 500 by 3600 (since there are 3600 seconds in an hour).
So, the number of vehicles per second is 500/3600 = 5/36.
Now, we need to find the headway in terms of the number of vehicles. To do this, we divide the headway in seconds (15 seconds) by the number of vehicles per second (5/36).
Headway in terms of vehicles = 15/(5/36) = 15 * (36/5) = 108.
Therefore, the headway in terms of vehicles is 108.
Now, substitute this value into the given equation P = 0.87t to find the probability:
P = 0.87 * 108 = 93.96.
Round this to two decimal places to get the final answer.
Therefore, the probability that the headway is at least 15 seconds is 93.96%, or approximately 0.94.